Let us first recall the six classes \(\mathcal {M}_=,\mathcal {M}_{\ne },\mathcal {M}_{\leqslant _R},\mathcal {M}_{\geqslant _R},\mathcal {M}_{\leqslant _C},\mathcal {M}_{\geqslant _C}\) 
of matrix obstructions to twin-width described in [1]}.The classes are called \(\mathcal {F}_=\) , etc. in [1]},
we use \(\mathcal {M}_=\)  to avoid ambiguity with tournaments defined in sec:obstructions.
Similarly to the classes of tournaments described in sec:obstructions,
each class \(\mathcal {M}_s\)  consists of the hereditary closure
of encodings \(\mathcal {M}_s(\sigma )\)  of arbitrary permutations \(\sigma \) .
If \(\sigma \)  is a permutation on \(n\)  elements, the \(n \times n\)  matrix
\(\mathcal {M}_s(\sigma ) = (m_{i,j})_{i,j \in [n]}\)  is described by
\(\mathcal {M}_=(\sigma ): && m_{i,j} = 1 & \iff j = \sigma (i) \\\mathcal {M}_{\ne }(\sigma ): && m_{i,j} = 1 & \iff j \ne \sigma (i) \\\mathcal {M}_{\leqslant _R}(\sigma ): && m_{i,j} = 1 & \iff j \leqslant \sigma (i) \\\mathcal {M}_{\geqslant _R}(\sigma ): && m_{i,j} = 1 & \iff j \geqslant \sigma (i) \\\mathcal {M}_{\leqslant _C}(\sigma ): && m_{i,j} = 1 & \iff i \leqslant \sigma ^{-1}(j) \\\mathcal {M}_{\geqslant _C}(\sigma ): && m_{i,j} = 1 & \iff i \geqslant \sigma ^{-1}(j)\) 
